If [x] denotes the greatest integer `le x`, then `[(2)/(3)]+[(2)/(3)+(1)/(99)]+[(2)/(3)+(2)/(99)\]+...+[(2)/(3)+(98)/(99)]` is equal to

Options

A. 99
B. 98
C. 66
D. 65

Explanation

To solve the expression \(\left[\frac{2}{3}\right] + \left[\frac{2}{3} + \frac{1}{99}\right] + \left[\frac{2}{3} + \frac{2}{99}\right] + \ldots + \left[\frac{2}{3} + \frac{98}{99}\right]\), we will analyze each term step by step.

### Step 1: Understand the Greatest Integer Function
The greatest integer function \([x]\) gives the largest integer less than or equal to \(x\). For example, \(\left[\frac{2}{3}\right] = 0\) because \(\frac{2}{3} \approx 0.666\) which is less than 1.

### Step 2: Calculate the First Term
The first term is:
\[ \left[\frac{2}{3}\right] = 0 \]

### Step 3: Analyze Subsequent Terms
Next, we need to analyze the terms \(\left[\frac{2}{3} + \frac{r}{99}\right]\) for \(r = 1, 2, \ldots, 98\).

### Step 4: Determine When the Value Changes
We need to find when \(\frac{2}{3} + \frac{r}{99} \geq 1\):
\[ \frac{2}{3} + \frac{r}{99} \geq 1 \]
Subtract \(\frac{2}{3}\) from both sides:
\[ \frac{r}{99} \geq 1 - \frac{2}{3} = \frac{1}{3} \]
Multiply both sides by 99:
\[ r \geq \frac{99}{3} = 33 \]

### Step 5: Calculate Terms for \(r < 33\)
For \(r = 1\) to \(32\):
\[ \left[\frac{2}{3} + \frac{r}{99}\right] = 0 \]
Thus, there are 32 terms contributing 0.

### Step 6: Calculate Terms for \(r \geq 33\)
For \(r = 33\) to \(98\):
\[ \left[\frac{2}{3} + \frac{r}{99}\right] = 1 \]
This is because for \(r = 33\), \(\frac{2}{3} + \frac{33}{99} = \frac{2}{3} + \frac{1}{3} = 1\) and for \(r\) values greater than 33, the sum remains greater than 1.

### Step 7: Count the Number of Terms from \(r = 33\) to \(r = 98\)
The number of terms from \(33\) to \(98\) is:
\[ 98 - 33 + 1 = 66 \]

### Step 8: Calculate the Total Sum
The total sum is:
\[ 0 \times 32 + 1 \times 66 = 66 \]

### Final Answer
Thus, the value of the given expression is:
\[ \boxed{66} \]